Coverage for local/lib/python2.7/site-packages/sage/categories/commutative_additive_groups.py : 33%

r""" 

Commutative additive groups 

""" 

#***************************************************************************** 

# Copyright (C) 2008 Teresa Gomez-Diaz (CNRS) <Teresa.Gomez-Diaz@univ-mlv.fr> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# http://www.gnu.org/licenses/ 

#****************************************************************************** 

from sage.categories.category_types import AbelianCategory 

from sage.categories.category_with_axiom import CategoryWithAxiom 

from sage.categories.algebra_functor import AlgebrasCategory 

from sage.categories.cartesian_product import CartesianProductsCategory 

from sage.categories.additive_groups import AdditiveGroups 

class CommutativeAdditiveGroups(CategoryWithAxiom, AbelianCategory): 

""" 

The category of abelian groups, i.e. additive abelian monoids 

where each element has an inverse. 

EXAMPLES:: 

sage: C = CommutativeAdditiveGroups(); C 

Category of commutative additive groups 

sage: C.super_categories() 

[Category of additive groups, Category of commutative additive monoids] 

sage: sorted(C.axioms()) 

['AdditiveAssociative', 'AdditiveCommutative', 'AdditiveInverse', 'AdditiveUnital'] 

sage: C is CommutativeAdditiveMonoids().AdditiveInverse() 

True 

sage: from sage.categories.additive_groups import AdditiveGroups 

sage: C is AdditiveGroups().AdditiveCommutative() 

True 

.. NOTE:: 

This category is currently empty. It's left there for backward 

compatibility and because it is likely to grow in the future. 

TESTS:: 

sage: TestSuite(CommutativeAdditiveGroups()).run() 

sage: sorted(CommutativeAdditiveGroups().CartesianProducts().axioms()) 

['AdditiveAssociative', 'AdditiveCommutative', 'AdditiveInverse', 'AdditiveUnital'] 

The empty covariant functorial construction category classes 

``CartesianProducts`` and ``Algebras`` are left here for the sake 

of nicer output since this is a commonly used category:: 

sage: CommutativeAdditiveGroups().CartesianProducts() 

Category of Cartesian products of commutative additive groups 

sage: CommutativeAdditiveGroups().Algebras(QQ) 

Category of commutative additive group algebras over Rational Field 

Also, it's likely that some code will end up there at some point. 

""" 

_base_category_class_and_axiom = (AdditiveGroups, "AdditiveCommutative") 

class CartesianProducts(CartesianProductsCategory): 

class ElementMethods: 

def additive_order(self): 

r""" 

Return the additive order of this element. 

EXAMPLES:: 

sage: G = cartesian_product([Zmod(3), Zmod(6), Zmod(5)]) 

sage: G((1,1,1)).additive_order() 

30 

sage: any((i * G((1,1,1))).is_zero() for i in range(1,30)) 

False 

sage: 30 * G((1,1,1)) 

(0, 0, 0) 

sage: G = cartesian_product([ZZ, ZZ]) 

sage: G((0,0)).additive_order() 

1 

sage: G((0,1)).additive_order() 

+Infinity 

sage: K = GF(9) 

sage: H = cartesian_product([cartesian_product([Zmod(2),Zmod(9)]), K]) 

sage: z = H(((1,2), K.gen())) 

sage: z.additive_order() 

18 

""" 

from sage.rings.infinity import Infinity 

orders = [x.additive_order() for x in self.cartesian_factors()] 

if any(o is Infinity for o in orders): 

return Infinity 

else: 

from sage.arith.functions import LCM_list 

return LCM_list(orders) 

class Algebras(AlgebrasCategory): 

pass